Tensors for Physics

102 7 Fields, Spatial Differential Operators

The plane wave is a special case of (7.63). In complex notation, this solution of

the wave equation reads

Eμ=Eμ(^0 )exp[ikνrν−iωt]. (7.64)

The wave vectorkνand the circular frequencyωare linked by thedispersion relation

kνkν=ω^2 /c^2 or

ω=kc, (7.65)

wherekis the magnitude of the wave vector.

7.2 Exercise: Test Solutions of the Wave Equation
Proof that both the ansatz (7.63) and the plane wave (7.64) obey the wave equation.
Furthermore, show that theE-field is perpendicular to the wave vector, and that the
B-field is perpendicular to both.

7.5.4 Scalar and Vector Potentials.

The electric fieldEand theB-field can be expressed as derivatives of the electroscalar

potentialφand a magnetic vector potentialAaccording to

Eμ=−∇μφ−

∂

∂t

Aμ, Bμ=εμνλ∇νAλ. (7.66)

With the ansatz (7.66), the homogeneous Maxwell equations (7.57) are fulfilled

automatically.

The electromagnetic potential functions, however, are not unique. More specifi-

cally, the same fieldsEandBfollow from (7.66), whenφandAare replaced by

φ′=φ−

∂

∂t

f, A′λ=Aλ+∇λf,

where f = f(t,r)is a scalar function. This allows, e.g. to requireφ= 0or

∇νAν=0.

For charges and currents in vacuum, whereD=ε 0 EandB=μ 0 H, insertion of

(7.66) into the inhomogeneous Maxwell equations (7.56) and use of the scaling

∂

∂t

φ+∇λAλ= 0 , (7.67)

leads to

Aν=ΔAν−

1

c^2

∂^2

∂t^2

Aν=μ 0 jν, φ=Δφ−

1

c^2

∂^2

∂t^2

φ=

1

ε 0

ρ. (7.68)